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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 8 — Aug. 1, 1999
  • pp: 2029–2044

Ellipsometric function of a film–substrate system: detailed analysis and closed-form inversion

A.-R. M. Zaghloul and M. S. A. Yousef  »View Author Affiliations

JOSA A, Vol. 16, Issue 8, pp. 2029-2044 (1999)

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The ellipsometric function ρ of a film–substrate system is analyzed through successive transformations from the plane of the two independent variables angle of incidence and film thickness (ϕd plane) to the complex ρ plane. This analysis is achieved by introducing two intermediate planes: the unimodular plane (Zi plane) and the translated ellipsometric plane (ρ* plane). The analysis through the Zi plane leads to classification of the film–substrate systems into two classes: clockwise and counterclockwise. The class of the film–substrate system governs the inversion from the ρ* plane to the Zi-plane. It identifies the number of branch points of ρ*-1 from the ρ* plane to the Zi plane. The branch points of ρ*-1 and its preimage in the ϕd plane are identified and studied. The domain of the double-valued function ρ*-1 is divided into two or four subdomains according to the class of the film–substrate system. In each of these subdomains, the single-valued branch of ρ*-1 is fixed, and we introduce a closed-form solution for the determination of the film thickness of the system. Mathematically, ρ*-1 exists in any domain that does not include the branch points. Hence the exceptive points are divided into two types: removable and essential. The closed-form inversion is obtained for the removable exceptive points. The conformality of both ρ and ρ*, as well as their inverses, leads to identification of the two essential exceptive inversion points, which exist at ϕ=0° and 90°. Accordingly, the closed-form solution is available throughout the ρ plane except at the two points ±1 (corresponding to ϕ=0° and 90°). A study of the extrema of the magnitude and the phase of both ρ and ρ* provides full information on the number of zeros and essential singularities for each of the three categories of film–substrate systems: negative, zero, and positive. Numerical examples are given to illustrate the introduced closed forms. Also, the table of transformation of regions between the ϕd plane and the ρ plane induced by ρ and ρ-1 is given.

© 1999 Optical Society of America

OCIS Codes
(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry
(120.4530) Instrumentation, measurement, and metrology : Optical constants
(120.5700) Instrumentation, measurement, and metrology : Reflection
(310.0310) Thin films : Thin films

Original Manuscript: November 5, 1998
Revised Manuscript: March 29, 1999
Manuscript Accepted: March 29, 1999
Published: August 1, 1999

A.-R. M. Zaghloul and M. S. A. Yousef, "Ellipsometric function of a film–substrate system: detailed analysis and closed-form inversion," J. Opt. Soc. Am. A 16, 2029-2044 (1999)

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  1. A.-R. M. Zaghloul, “A study of the ellipsometric function of a film–substrate system and applications to the design of reflection-type optical devices and to ellipsometry,” Ph.D. dissertation (University of Nebraska–Lincoln, Lincoln, Nebraska, 1975).
  2. M. M. Ibrahim, N. M. Bashara, “Parameter-correlation and computational considerations in multiple-angle ellipsometry,” J. Opt. Soc. Am. 61, 1622–1629 (1971). [CrossRef]
  3. J. Lekner, “Analytic inversion of ellipsometric data for an unsupported nonabsorbing uniform layer,” J. Opt. Soc. Am. A 7, 1875–1877 (1990). [CrossRef]
  4. D. U. Fluckiger, “Analytic methods in the determination of optical properties by spectral ellipsometry,” J. Opt. Soc. Am. A 15, 2228–2232 (1998). [CrossRef]
  5. V. G. Polovinkin, S. N. Svitasheva, “Analysis of general ambiguity of inverse ellipsometric problem,” Thin Solid Films 313, 128–131 (1998). [CrossRef]
  6. M. Ghezzo, “Thickness calculation for transparent film from ellipsometric measurements,” J. Opt. Soc. Am. 58, 368–372 (1968). [CrossRef]
  7. R. M. A. Azzam, “Simple and direct determination of complex refractive index and thickness of unsupported or embedded thin films by combined reflection and transmission ellipsometry at 45° angle of incidence,” J. Opt. Soc. Am. 73, 1080–1082 (1983). [CrossRef]
  8. C. G. Galarza, P. P. Khargonekar, N. Layadi, T. L. Vincent, E. A. Rietman, J. T. C. Lee, “A new algorithm for real-time thin-film thickness estimation given in-situ multiwavelength ellipsometry using an extended Kalman filter,” Thin Solid Films 313, 156–160 (1998). [CrossRef]
  9. M. S. A. Yousef, A.-R. M. Zaghloul, “Ellipsometric function of a film–substrate system: characterization and detailed study,” J. Opt. Soc. Am. A 6, 355–366 (1989). [CrossRef]
  10. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Ellipsometric function of a film–substrate system: applications to the design of reflection-type optical devices and to ellipsometry,” J. Opt. Soc. Am. 65, 252–260 (1975). [CrossRef]
  11. The two bilinear transformations of Eqs. (6) and (7) are obtained by translating the centers of the two circles AZ and B/Z of Ref. 9 into the origin of the complex plane.
  12. R. M. A. Azzam, M. E. R. Khan, “Complex reflection coefficients for parallel and perpendicular polarizations of a film–substrate system,” Appl. Opt. 22, 253–264 (1983). [CrossRef] [PubMed]
  13. Multiple-valued functions may have singular points that are not removable singularities, poles, or essential singularities. These singular points are called branch points; more precisely, they are points at which the double-valued function becomes a single-valued one.
  14. The two values of ϕ=0° and 90° are deleted from the domain of the angle of incidence because the translated-ellipsometric function ρ* at both values of ϕ is not a conformal mapping (see Subsection 3.C).
  15. The pair of equations that give the same value of x are not the same. The parameters of one of these equations depend on only one of the two fundamental polarizations, either parallel or perpendicular. Those of the second equation depend only on the other fundamental polarization.
  16. From the definition of ln z, where z is a complex variable, we have ln z=∫γ drr+j∫γ dθ=ln|z|+jΔγ arg z. That is, the value of ln z depends not only on the point z but also on the curve γ along which the integral is taken. To overcome the troublesome determination of the curve γ to obtain the closed forms of dr, we use the Cartesian representation of the point x(ρ*).
  17. A.-R. M. Zaghloul, R. M. A. Azzam, N. M. Bashara, “Inversion of the nonlinear equations of reflection ellipsometry on film–substrate systems,” Surf. Sci. 5, 87–96 (1976). [CrossRef]
  18. The polarizer–surface-analyzer null ellipsometry is to be used experimentally to determine the number of the real-axis crossing by ρ for the three categories of the film–substrate system; see Ref. 19. The very simple single-element rotating-polarizer ellipsometry may be used for only the zero and the negative systems; see A.-R. M. Zaghloul, R. M. A. Azzam, “Single-element rotating-polarizer ellipsometer for film–substrate systems,” J. Opt. Soc. Am. 67, 1286–1287 (1977). [CrossRef]
  19. R. M. A. Azzam, A.-R. M. Zaghloul, N. M. Bashara, “Polarizer–surface-analyzer null ellipsometry for film–substrate systems,” J. Opt. Soc. Am. 65, 1464–1471 (1975). [CrossRef]

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